Spontaneous symmetry breaking -
As mentioned above, the ansatz (REF ) for the HFB wave function
implies that this wave function is not an eigenstate of the particle number
operator \(\hat{N} = \sum _{i} c_i^{\dagger }c_i\) . Similarly, solving the HFB
equation (REF ) with constraints on the expectation value of multipole
moments in the intrinsic frame implies that the density matrix \(\rho \)  is not
going to be rotationally invariant. Two fundamental symmetries of the nuclear
Hamiltonian, particle number and rotational invariance, are thus broken
in the HFB theory. This spontaneous symmetry breaking is a trademark of the
EDF theory and one of the main reasons why, in practice, this approach provides
an accurate model for atomic nuclei. As discussed extensively in [1]}, [2]}, [3]}, the symmetry group \(\mathcal {G}\)  of the nuclear
Hamiltonian reads
\(\mathcal {G} = T(3) \times SO(3) \times \Pi \times U(1)_{N} \times U(1)_{Z} \times SU(2)\) 
